I once cried just like that at about his age when my dad took me to Old Trafford to watch a Derby whilst sat in the united end. At that point, I think he was still holding out some hope that I’d capitulate and start following united. Ever so glad he called that one wrong.
It was a midweek fixture, in or around 1978, the original game having been postponed due to a waterlogged pitch. United went 2-0 up and I burst into tears, only for City to pull two goals back (Kidd and Barnes iirc) to draw the game. I wasn’t crying at the end.
Good luck to the kid. Nothing wrong with caring about your football club, whatever your age.